Friday, February 16, 2018 - 15:00 , Location: Skiles 271 , Yian Yao , GT Math , Organizer: Jiaqi Yang
I will report on the parameterization method for computing normally hyperbolic invariant tori(NHIT) for diffeomorphisms. To this end, a Newton-like method for solving the invariance equation based on the graph transform method will be presented with details. Some notes on numerical implementations will also be included if time allows. This is a work by Marta Canadell and Alex Haro.
Friday, February 9, 2018 - 15:00 , Location: Skiles 271 , Joan Gimeno , BGSMath-UB , Organizer: Jiaqi Yang
We are going to explain how invariant dynamical objects, such as (quasi)periodic orbits, can numerically be computed for Delay Differential Equations as well as their stability. To this end, we will use Automatic Differentiation techniques and iterative linear solvers with appropiate preconditioners. Additionally some numerical experiments will be presented to illustrate the approaches for each of those objects.This is joint work with A. Jorba.
Friday, February 2, 2018 - 15:00 , Location: Skiles 271 , Gladston Duarte , University of Barcelona & GT , firstname.lastname@example.org , Organizer: Jiaqi Yang
In a given system of coordinates, the Restricted Three-Body Problem has some interesting dynamical objects, for instance, equilibrium points, periodic orbits, etc. In this work, some connections between the stable and unstable manifolds of periodic orbits of this system are studied. Such connections let one explain the movement of Quasi-Hilda comets, which describe an orbit that sometimes can be approximated by an ellipse of semi-major axis greater than Jupiter's one, sometimes smaller. Using a computer algebra system, one can compute an approximation to those orbits and its manifolds and investigate the above mentioned connections. In addition, the Planar Circular model is used as a base for the fitting of the orbit of comet 39P/Oterma, whose data were collected from the JPL Horizons system. The possibility of using other models is also discussed.
Friday, November 17, 2017 - 15:00 , Location: Skiles 154 , Bhanu Kumar , GT Math , Organizer:
This lecture will discuss the stability of perturbations of integrable Hamiltonian systems. A brief discussion of history, integrability, and the Poincaré nonintegrability theorem will be followed by the proof of the theorem of Kolmogorov on persistence of invariant tori. Time permitting, the problem of small divisors may be briefly discussed. This lecture wIll follow the slides from the Satellite Dynamics and Space Missions 2017 summer school held earlier this semester in Viterbo, Italy.
Friday, November 10, 2017 - 14:00 , Location: Skiles 154 , Rafael de la Llave , GT Math , Organizer: Jiaqi Yang
We consider Hamiltonian systems with normally hyperbolic manifold with a homoclinic connection. The systems are of the form H_0(I, phi, x,y) = h(I) + P(x,y) ,where P is a one dimensional system with a homoclinic intersection. The above Hamiltonian is a standard normal form for near integrable Hamiltonians close to a resonance. We consider perturbations that are time dependent and may be not Hamiltonian. We derive explicit formulas for the first order effects on the stable/unstable manifolds. In particular, we give sufficient conditions for the existence of homoclinic intersections to the normally hyperbolic manifold. Previous treatments in the literature specify the types of the unperturbed orbits considered (periodic or quasiperiodic) and are restricted to periodic or quasi-periodic perturbations. We do not need to distinguish on the perturbed orbits and we allow rather general dependence on the time (periodic, quasiperiodic or random). The effects are expressed by very fast converging improper integrals. This is joint work with M. Gidea. https://arxiv.org/abs/1710.01849
Friday, November 3, 2017 - 15:00 , Location: Skiles 154 , Hassan Attarchi , Georgia Tech , Organizer:
This presentation is about the results of a paper by L. Bunimovich in 1974. One considers dynamical systems generated by billiards which are perturbations of dispersing billiards. It was shown that such dynamical systems are systems of A. N. Kolmogorov (K-systems), if the perturbation satisfies certain conditions which have an intuitive geometric interpretation.
Friday, October 27, 2017 - 15:00 , Location: Skiles 154 , Hassan Attarchi , Georgia Tech , Organizer:
This presentation is about the results of a paper by Y. Sinai in 1970. Here, I will talk about dynamical systems which resulting from the motion of a material point in domains with strictly convex boundary, that is, such that the operator of the second quadratic form is negative-definite at each point of the boundary, where the boundary is taken to be equipped with the field of inward normals. It was proved that such systems are ergodic and are K-systems. The basic method of investigation is the construction of transversal foliations for such systems and the study of their properties.
Friday, October 13, 2017 - 15:00 , Location: Skiles 154 , Bhanu Kumar , GT Math , Organizer: Jiaqi Yang
Birkhoff's Theorem is a result useful in characterizing the boundary of certain open sets U ⊂ T^1 x [0, inf) which are invariant under "vertical-tilting" homeomorphisms H. We present the method used by A. Fathi to prove Birkhoff's theorem, which develops a series of lemmas using topological arguments to prove that this boundary is a graph.
An infinite dimensional hamiltonian dynamical system from MFG theory II (CANCELED, SEE SUBSTITUTE TALK)Friday, October 6, 2017 - 15:00 , Location: Skiles 154 , Sergio Mayorga , Georgia Tech , Organizer: Jiaqi Yang
We will look at a system of hamiltonian equations on the torus, with an initial condition in momentum and a terminal condition in position, that arises in mean field game theory. Existence of and uniqueness of solutions will be shown, and a few remarks will be made in regard to its connection to the minimization problem of a cost functional. This is the second part of lasrt week's talk.
Friday, October 6, 2017 - 15:00 , Location: Skiles 154 , Prof. Rafael de la Llave , School of Mathematics, Georgia Tech , Organizer: Jiaqi Yang
We will present an introduction to the results of S. Aubry and J. Mather who used variational methods to prove the existence of quasi-periodic orbits in twist mappings and in some models appearing in solid state Physics.